Branching processes in random environment

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Branching Processes

in Random Environment

Dissertation

zur Erlangung des Doktorgrades

der Naturwissenschaften

vorgelegt beim Fachbereich 12, Informatik und Mathematik der Johann Wolfgang Goethe-Universit¨at

in Frankfurt am Main

von

Christian B¨oinghoff aus Erlenbach am Main

200 400 600 800 1000 0 2 4 6 8 Branching Process Z lo g10 (( Z )) n Frankfurt am Main 2010 (D30)

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vom Fachbereich 12, Informatik und Mathematik der

Johann Wolfgang Goethe-Universit¨at als Dissertation angenommen.

Dekan: Prof. Dr. Tobias Weth

Gutachter: Prof. Dr. G¨otz Kersting , Prof. Dr. Anton Wakolbinger und Prof. Dr. Nina Gantert

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The following PhD-thesis emerged from a joint project with scientists from Germany, Russia and France. I would also like to express my gratitude to the German Research Foundation (DFG) and the Russian Foundation of Basic Research for financial support (Grant DFG-RFBR 08-01-91954). It is a pleasure to express gratitude to all people involved in different parts of the project.

First of all, I am indebted to G¨otz Kersting1 _{who supervised this PhD, provided many helpful ideas and}

always had time to discuss questions with me. His guidance into the world of branching processes has been essential for this thesis.

Secondly, I am very grateful to Vincent Bansaye2 for the fruitful and enduring collaboration, as well as for invitations to the ´Ecole Polytechnique and his cordial hospitality.

On the Russian side, I would like to thank Elena Dyakonova, Vladimir Vatutin and Valery Afanasyev3

for the good cooperation, many fruitful discussions, the invitations to the Steklov Institute in Moscow, and in particular for their extraordinary hospitality during two stays in Moscow.

I am also grateful to Vitali Wachtel4 _{for many interesting discussions on BPREs, as well as for quite}

practical help with language issues during the time in Moscow.

I would also like to thank Nina Gantert5_{for fruitful discussions on the large deviations of BPREs in the}

quenched approach, as well as for the invitation to M¨unster.

During my work in Frankfurt, many people helped me with mathematical, technical or organzational questions. Among them, I’d especially like to thank our two secretaries, Nicole G¨otting and Anna Weigl-hofer for their technical support. I’d also like to thank the professors Hermann Dinges, Ralph Neininger, Gaby Schneider and Anton Wakolbinger for many interesting discussions, especially in the vivid stochas-tic seminary. I’d also like to thank the other PhD students from the stochasstochas-tic group for the interesting exchange of ideas and the good working atmosphere: Margarete Knape, Max Stroh, Henning Sulzbach and, from statistics, Markus Bingmer who provided very valuable help with a few R problems. I am grateful to Brooks Ferebee for many useful remarks after talks in our seminary and for providing support with subtleties of the English language.

I’d also like to thank my family for their enduring support, especially Andreas for carefully looking for any typos in my PhD-thesis and Ute for carefully reading the pre-final version of the thesis, her advice and many helpful remarks.

1_{Goethe University, Frankfurt/Main} 2_{Ecole Polytechnique, Palaiseau}´

3_{all from Steklov Institute, Moscow} 4_{LMU, M¨}_unchen

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Zusammenfassung V

Zusammenfassung

In der folgenden Arbeit werden Eigenschaften von Verzweigungsprozessen in zufälliger Umgebung (engl. Branching processes in random environment, kurz BPREs) untersucht. Das Modell geht auf [SW69] und [AK71] zurück. Ein BPRE ist ein einfaches mathematisches Modell für die Entwicklung einer Population von apomiktischen6 _{Individuen in diskreter Zeit, wobei die Umgebungsbedingungen}

einen Einfluß auf den Fortpflanzungserfolg der Individuen haben. Dabei wird angenommen, dass die Umgebungsbedingungen in den einzelnen Generationen zufällig sind, und zwar unabhängig und identisch verteilt von Generation zu Generation. Man denke z.B. an eine Population von Pflanzen mit einem einjährigen Zyklus, die in jedem Jahr anderen Witterungsbedingungen ausgesetzt sind, wobei angenom-men wird, dass diese sich unabhängig und identisch verteilt ändern.

Genauer bezeichnen wir eine unendliche Folge von unabh¨angig, identisch verteilten Zufallsvariablen Q1, Q2, . . ., die Werte im Raum ∆ aller Wahrscheinlichkeitsverteilungen auf N0annehmen, als eine

Umge-bung Π = (Q1, Q2, . . .). Ein BPRE wird dann wie folgt definiert:

Definition. Sei Π = (Q1, Q2, . . .). Dann bezeichnen wir (Zn)n∈N0 als Verzweigungsprozess in

zuf¨alliger Umgebung, falls f¨_{ur alle z, k ∈ N}0 die Populationsgr¨oße Zk in Generation k, gegeben

Zk−1 = z und gegeben Π = (q1, q2, . . .), wie die Summe von z-vielen unabh¨angig, identisch verteilten

Zufallsvariablen verteilt ist, d.h.:

L(Zk|Zk−1= z, Π = (q1, q2, · · · )) = L(ξ1+ · · · + ξz) ,

wobei ξ1, ξ2, . . . , ξz unabh¨angige Zufallsvariablen mit Verteilung qk−1 sind.

Als Hilfsmittel definiert man die zugeh¨orige Irrfahrt. Definition. Seien Π = (Q1, Q1, . . .) eine Umgebung und

Xn := log ∞

X

y=0

yQn({y}), n ≥ 1 .

Die Irrfahrt S = (S0, S1, . . .) mit Anfangszustand S0= 0 und Zuw¨achsen Xn= Sn− Sn−1, n ≥ 1, heißt

zugeh¨orige Irrfahrt f¨ur den Prozess (Zn)n∈N0.

Die zugeh¨orige Irrfahrt bestimmt den Erwartungswert des Prozesses, bedingt auf die Umgebung: E[Zn|Z0, Π] = Z0 eSn f.s.

Verzweigungsprozesse in zufälliger Umgebung werden, ähnlich wie gewöhnliche Galton-Watson Prozesse, in superkritische (E[X] > 0), kritische (E[X] = 0) und subkritische Prozesse (E[X] < 0) unterteilt. Kri-tische und subkriKri-tische Prozesse sterben f.s. aus (siehe Kapitel 1).

Bereits in den Arbeiten [Koz76] und [Afa80] wird ein interessantes Verhalten von BPREs im subkritischen Fall beschrieben, zunächst jedoch nur für Nachkommenverteilungen mit gebrochen-linearen Erzeugenden-funktionen. In diesem Fall lässt sich die Erzeugendenfunktion von Zn, bedingt auf die Umgebung, explizit

berechnen. Im subkritischen Fall gibt es drei verschiedene Regime von Verzweigungsprozessen, die sich in der Asymptotik der Überlebenswahrscheinlichkeit und im Verhalten des Prozesses, bedingt auf Überleben (d.h. bedingt auf {Zn > 0}, n ∈ N), unterscheiden. In späteren Arbeiten, z.B. [GKV03], [AGKV05b],

[AGKV05a] und [ABKV10] wird dies detailliert beschrieben und unter schwachen Voraussetzungen an die Nachkommenverteilungen und die Verteilung von X gezeigt. Man unterscheidet den schwach sub-kritischen (E[XeX_{] > 0), den intermedi¨}

ar subkritischen (E[XeX_{] = 0) und den stark subkritischen Fall}

(E[XeX_{] < 0). Einige bekannte Resultate der letzten Jahre werden in Kapitel 2 vorgestellt.}

In [AGKV05b] wird gezeigt, dass Z im stark subkritischen Fall, bedingt auf Überleben, zu allen Zeiten klein bleibt und im Grenzwert als Markovkette beschrieben werden kann. Die Überlebenswahrscheinlichkeit P(Zn> 0) fällt exponentiell schnell ab, und zwar mit Ordnung E[eX]n.

Der schwach subkritische Fall wird in [ABKV10] beschrieben. Auch in diesem Fall f¨_{allt P(Z}n > 0)

ex-ponentiell schnell ab, ist aber von derselben Ordnung wie P(min{S0, . . . , Sn} ≥ 0). Unter geeigneten

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VI Zusammenfassung Voraussetzungen konvergiert der mit dem Erwartungswert skalierte Verzweigungsprozess, bedingt auf

¨

Uberleben, in Verteilung gegen eine Exkursion einer Brown’schen Bewegung. Dies bedeutet, dass Zk,

bedingt auf {Zn > 0}, sowohl für k nahe Null als auch für k nahe n beschränkt bleibt. Dazwischen

nimmt Zk sehr große Werte an und folgt seinem Erwartungswert, bis auf eine zuf¨allige Konstante, auf

völlig deterministische Art und Weise. Hier kommt das starke Gesetz der großen Zahl zum Tragen. Die Untersuchung von intermediär subkritischen Verzweigungsprozessen, bedingt auf Überleben, ist einer der Hauptteile dieser Dissertation. Aufgrund der Bedingung

E[XeX] = 0

bietet es sich für die Beweise an, einen Maßwechsel durchzuführen, unter welchem die zugehörige Irrfahrt rekurrent wird. Hierzu definieren wir das Maß P mit Erwartungswert E für messbare und beschränkte Funktionen Φ : ∆n × Rn → R durch E [Φ(Q1, . . . , Qn, Z0, . . . , Zn)] = γ−nEΦ(Q1, . . . , Qn, Z0, . . . , Zn)eSn−S0 mit γ = _EeX .

S ist unter P rekurrent, d.h. E[X] = 0. Wir nehmen an, dass die Verteilung von X folgende Regu-larit¨atsbedingung erf¨ullt:

Annahme 0.1. Die Verteilung von X hat bzgl. P endliche Varianz oder, allgemeiner, liegt im Konver-genzbereich einer strikt stabilen Verteilung mit Index α ∈ (1, 2]. Zudem sei sie nicht-gitterartig.

Außerdem muss eine gewisse Regularit¨at der Nachkommenverteilungen vorausgesetzt werden. Sie betrifft das sogenannte standardisierte zweite Moment von Q. Dazu definiert man

ζ(a) = ∞ X y=a y2Q({y}).m(Q)2, _{a ∈ N .} mit m(Q) =P∞ y=0yQ({y}).

Annahme 0.2. Es existieren Konstanten 0 < < ∞ und a ∈ N, so dass E[(log+ζ(a))α+] < ∞ .

Wie in der Arbeit später ausführlich erklärt wird, erfüllt eine große Klasse von Verteilungen die obige Bedingung.

Sei

τn := min0 ≤ k ≤ n|Sk= min{S0, . . . , Sn}

(1) der Zeitpunkt des ersten Minimums von (S0, . . . , Sn).

Der folgende Satz wird bereits in [Vat04] gezeigt, jedoch unter etwas st¨arkeren Voraussetzungen. Satz 0.1. Unter den Annahmen 0.1 und 0.2 gilt

P(Zn> 0) ∼ γnθP(τn= n)

f¨ur ein 0 < θ < ∞.

Der n¨achste Satz beschreibt die Anzahl der Zeiten – bedingt aufs ¨Uberleben des Prozesses – zu denen nur noch genau ein Individuum lebt. Dazu nehmen wir an, dass es mit positiver Wahrscheinlichkeit Verteilungen gibt, unter welchen ein Individuum mit positiver Wahrscheinlichkeit keine oder genau einen Nachkommen haben kann:

Annahme 0.3.

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Zusammenfassung VII Die Anzahl der Zeitpunkte, zu denen nur noch genau ein Individuum lebt ist dann von der gleichen Ordnung wie die Anzahl der strikt absteigenden Leiterpunkte einer rekurrenten Irrfahrt, bedingt auf {τn= n}.

Satz 0.2. Unter den Annahmen 0.1 bis 0.3 gibt es eine schwach variierende Folge b1, b2, . . ., so dass

E h ]{k|Zk = 1} Zn > 0 i = Θ(bn n1−ρ) gilt.

Dabei bezeichnet xn= Θ(yn), dass die Folge xnf¨ur n → ∞ von der gleichen Ordnung wie ynist. Genauer

gesagt gibt es Konstanten c1, c2∈ R+, so dass

c1 ≤ lim inf n→∞ xn yn ≤ lim sup n→∞ xn yn ≤ c2 .

Wie bereits durch Satz 0.1 angedeutet wird, konvergiert die zugehörige Irrfahrt, bedingt aufs Überleben des Prozesses, gegen einen Lévy-Prozess, der darauf bedingt ist, sein Minimum am Ende anzunehmen. Satz 0.3. Unter Annahmen 0.1 und 0.2 gilt für n → ∞,

L Sbntc/abntc 0≤t≤1 Zn> 0 _d → L(L−) ,

in Verteilung bzgl. der Skorohod Metrik, wobei L− einen L´evy-Prozess bezeichnet, der darauf bedingt ist, sein Minimum am Ende anzunehmen.

Im intermediär subkritischen Fall zeichnet sich also folgendes Bild ab: Der Prozess überlebt typischer-weise in Umgebungen in denen sich die zugehörige Irrfahrt wie eine rekurrente Irrfahrt verhält, die darauf bedingt ist, ihr Minimum am Ende anzunehmen. Das heißt, dass Überleben nicht allein dadurch real-isiert wird, dass der Prozess einer außergewöhnlich ‘günstigen’ Umgebung7_{ausgesetzt ist (wie im schwach}

subkritischen Fall) sondern durch ungew¨ohnlich hohe Nachkommenzahlen innerhalb einer ‘ung¨unstigen’ Umgebung8. Eine rekurrente Irrfahrt, bedingt darauf ihr Minimum am Ende anzunehmen, kann lange Exkursionen zwischen den absteigenden Leiterpunkten besitzen. In diesen Perioden folgt der Prozess seinem Erwartungswert.

Im zweiten Teil der Arbeit werden große Abweichungen behandelt. Die Wahrscheinlichkeit, dass der Prozess außergew¨ohlich große Werte annimmt, f¨allt exponentiell schnell ab. In dieser Arbeit wird die Ratenfunktion ψ : R → R+_{∪ {∞} bestimmt, die}

P(Zn> eθn) = e−ψ(θ)n+o(n)

erfüllt. Für unsere Untersuchung nehmen wir an, dass eine nichtentartete Ratenfunktion für die zugehörige Irrfahrt existiert, was durch die sogenannte rechtsseitige Cramér-Bedingung sichergestellt wird. Es wird gefordert, dass die zugehörige Irrfahrt endliche exponentielle Momente besitzt:

Annahme 0.4. Es existiert ein s > 0, so dass die momentenerzeugende Funktion endlich ist: ϕ(s) := EesX < ∞ .

Insbesondere existiert E[X] ≥ −∞.

Für subkritische BPREs ist bereits das Überleben des Prozesses ein Ereignis mit exponentiell schnell abfallender Wahrscheinlichkeit. Ein einfaches Subadditivitätsargument liefert allgemein die Existenz des Grenzwertes

γ := lim

n→∞−

1

nlog P(Zn> 0) ,

7_‘g¨_{unstige’ Umgebung bedeutet min{S}

0, · · · , Sn} ≥ 0 8_‘ung¨_{unstige’ Umgebung bedeutet, dass das min{S}

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VIII Zusammenfassung wobei 0 ≤ γ < ∞.

Sei Λ die Ratenfunktion der zugeh¨origen Irrfahrt, d.h.

P(Sn ≥ θn) = e−Λ(θ)n+o(n)

mit

Λ(θ) := sup

s≥0

sθ − log ϕ(s) .

Wie sich herausstellt, h¨angt f¨ur eine große Klasse von Nachkommenverteilungen (diejenigen, bei denen alle Momente endlich sind) die Ratenfunktion ψ nur von γ und Λ ab.

Von Bedeutung ist die Funktion Γ : R+→ R ∪ {∞}. F¨ur alle θ ≥ 0 ist Γ durch Γ := inf

0<t≤1tγ + (1 − t)Λ(θ/(1 − t)) (2)

definiert.

In Kapitel 4.2 wird zun¨achst der Fall von Nachkommenverteilungen behandelt, deren Tails geometrisch schnell abfallen, was durch folgende Annahme sichergestellt wird.

Annahme 0.5. Es existieren Konstanten k0 ∈ N0, 0 ≤ a < b und c > 0, so dass Q f.s. Werte in der

Menge aller Wahrscheinlichkeitsverteilungen A ⊂ ∆ mit der folgenden Eigenschaft annimmt: Falls R Verteilung P und Erwartungswert E [R] = m hat, so gilt

E(R − j)+

≤ c m a + m b + m

j−k0

, j ≥ k0. (3)

Unter dieser Voraussetzung gilt ψ = Γ, d.h.

Satz 0.4. Unter Annahmen 0.4 und 0.5 gilt f¨ur jedes θ ≥ 0 lim sup n→∞ 1 nlog P(Zn≥ e θn₎ _≤ _−Γ(θ) _, lim inf n→∞ 1 nlog P(Zn> e θn₎ _≥ _{−Γ(θ+) .}

In Kapitel 4.3 werden Nachkommenverteilungen mit schweren Tails behandelt.

Die folgende Annahme stellt sicher, dass die Tails der Nachkommenverteilungen, gleichm¨aßig ¨uber alle Umgebungen, mindestens mit Exponent β ∈ (1, ∞) abfallen:

Annahme 0.6. Es existiert eine Konstante 0 < d < ∞, so dass Q f.s. Werte im Raum der Wahrschein-lichkeitsverteilungen A ⊂ ∆ mit der folgenden Eigenschaft annimmt:

Falls R Verteilung P und Erwartungswert E [R] = m hat, so gilt f¨ur alle z > 0 P(R > z |R > 0) ≤ d (m ∧ 1) z−β f.s.

Die Ratenfunktion ψ h¨angt dann nicht nur von γ und Λ ab, sondern auch von β. Sie ist durch ψ(θ) = ψγ,β,Λ(θ) := inf

t∈[0,1],s∈[0,θ]

n

tγ + βs + (1 − t)Λ((θ − s)/(1 − t))o (4) definiert.

In Kapitel 4.3 wird dann folgender Satz gezeigt:

Satz 0.5. Falls ein β ∈ (1, ∞) existiert mit log(P(Z1> z))/ log(z) z→∞

−→ −β und zusätzlich Annahme 0.6 für dieses β erfüllt ist, so gilt für jedes θ ≥ 0

1

nlog(P(Zn≥ e

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Zusammenfassung IX Bemerkung. Die erste Annahme in Satz 0.5 stellt sicher, dass mit positiver Wahrscheinlichkeit Nachommenverteilungen mit schweren Tails auftreten, deren Tails mit Exponent β abfallen.

Mithilfe des obigen Satzes l¨asst sich Satz 0.4 leicht verallgemeinern:

Satz 0.6. Falls Annahme 0.6 für jedes β > 0 erfüllt ist, so gilt für jedes θ ≥ 0 lim sup n→∞ 1 nlog P(Zn ≥ e θn₎ _≤ _−Γ(θ) _, lim inf n→∞ 1 nlog P(Zn > e θn₎ _≥ _{−Γ(θ+) ,} wobei Γ(θ) = inft∈[0,1]{tγ + (1 − t)Λ(θ/(1 − t))}.

In Kapitel 4.3 werden Ereignisse untersucht, deren Wahrscheinlichkeiten exponentiell schnell abfallen und zwar mit einer Rate, die durch die Ratenfunktion ψ bestimmt wird. Diese Ereignisse können auf ver-schiedenen Wegen realisiert werden. Der exponentielle Abfall wird jedoch durch den ‘günstigsten’ Weg bestimmt. Im Fall von Verteilungen mit schweren Tails und für große θ besteht diese ‘optimale Strategie’ daraus, dass zunächst ein Individuum exponentiell viele Nachkommen hat und der Prozess anschließend in einer günstigen Umgebung (d.h. hier, dass die zugehörige Irrfahrt linear wächst) gemäß seinem Er-wartungswert wächst.

Im stark subkritischen Fall und für kleine θ besteht die optimale Strategie daraus, zunächst in einer ungünstigen Umgebung (d.h. hier, dass die zugehörige Irrfahrt linear fällt) bis zur Zeit btnc, t ∈ (0, 1) nur zu überleben. Der Prozess überlebt zwar, bleibt aber beschränkt. Erst ab dem Zeitpunkt btnc wird eine günstige Umgebung realisiert und der Prozess wächst entsprechend seinem Erwartungswert. Dieser Effekt wird, ebenso wie Sonderfälle, detailliert in Kapitel 4.3.1 beschrieben.

Die Beweise von Satz 0.4 und 0.5 beruhen darauf, als untere Schranke die Wahrscheinlichkeiten entlang einer ‘optimalen Strategie’ zu maximieren. Für die obere Schranke wird eine Abschätzung der Tail-wahrscheinlichkeiten von Zn, bedingt auf die Umgebung, benötigt. Diese Abschätzung erhält man über

die Berechnung und Absch¨atzung von Ableitungen von Erzeugendenfunktionen.

In Kapitel 4.4 wird die exponetielle Abfallrate der Wahrscheinlichkeit P(0 ≤ Zn ≤ eθn) f¨ur superkritische

BPREs (E[X] > 0) im Fall von Nachkommenverteilungen mit gebrochen-linearer Erzeugendenfunktion bestimmt. In diesem Fall kann die Verteilung von Zn, bedingt auf die Umgebung, explizit berechnet

werden.

Annahme 0.7. Q nimmt P– f.s. Werte in der Menge aller Wahrscheinlichkeitsverteilungen A ⊂ ∆ mit der folgenden Eigenschaft an:

Sei R eine Zufallsvariable mit Verteilung P. Dann ist f¨ur s ∈ [0, 1] die Erzeugendenfunktion durch fR(s) := E [sR] = 1 −

1 − s

m−1_R + 1/2 bRm−2R (1 − s)

gegeben, mit mR=P∞k=0kP(R = k) und bR=P∞k=0k(k −1)P(R = k). Zus¨atzlich existieren Konstanten

0 < c1< c2< ∞ (gleichm¨aßig f¨ur alle P) so dass

c1 < 1/2 bRm−2R < c2 .

Zur Vereinfachung wird angenommen, dass die momentenerzeugende Funktion der Zuw¨achse der Irrfahrt f¨_{ur alle s ∈ R endlich ist,}

ϕ(s) := E[esX] < ∞ f¨ur alle s ∈ R und ung¨unstige Umgebungen auftreten k¨_{onnen, d.h. P(X < 0) > 0.}

Definiert man die exponentielle Abfallrate der Wahrscheinlichkeit, dass der Prozess ¨uberlebt, aber beschr¨ankt bleibt,

% := − lim

n→∞

1

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X Zusammenfassung so ist die gesuchte Ratenfunktionn χ durch

χ(θ) := inf

t∈[0,1)t% + (1 − t)Λ(θ/(1 − t))

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Satz 0.7. Es sei E[X] > 0 und Annahme 0.7 erf¨ullt. Dann gilt f¨_{ur alle 0 < θ < E[X]} lim n→∞ 1 nlog P(1 ≤ Zn≤ e θn₎ ₌ −χ(θ) und f¨_{ur alle θ ≥ E[X],}

lim n→∞ 1 nlog P(Zn≥ e θn₎ ₌ −χ(θ) (= Λ(θ)) .

F¨ur kleine θ besteht nun die optimale Strategie, zu ¨Uberleben und kleiner als eθn _{zu bleiben darin, bis}

zur Zeit btnc innerhalb einer günstigen Umgebung beschränkt zu bleiben. Erst anschließend wächst der Prozess entsprechend seinem Erwartungswert.

In Kapitel 4.5 werden große Abweichungen, bedingt auf die Umgebung (engl. quenched), untersucht. Außergewöhnlich große Werte können hier nicht mehr über die Umgebung realisiert werden. Für Nachkom-menverteilungen mit schweren Tails gilt:

Satz 0.8. Falls lim supz→∞log P(Z1 > z|Π, Z0 = 1)/ log z = −β f.s. f¨ur ein β ∈ (1, ∞) und Annahme

0.6 für dieses β erfüllt ist, so gilt für jedes θ ≥ (E[X] ∨ 0), lim n→∞ 1 nlog P(Zn> e θn_|Π) ₌

−β(θ − E[X]) , falls E[X] > 0 −(βθ − E[X]) , falls E[X] ≤ 0 f.s.

Falls alle Nachkommenverteilungen geometrisch beschränkte Tailwahrscheinlichkeiten besitzen, so gilt: Satz 0.9. Falls P(Z1> eθ|Π) > 0 f.s. und Annahme 0.5 erfüllt ist, so gilt für jedes θ ≥ (E[X] ∨ 0)

lim n→∞ 1 nlog(− log P(Zn> e θn_|Π)) ₌

θ − E[X] , falls E[X] > 0

θ , _{falls E[X] ≤ 0} f.s.

Die Wahrscheinlichkeit, außergew¨ohnlich große Werte zu realisieren, ist von kleinerer als exponentieller Ordnung.

Zum Abschluss der Dissertation werden Verzeigungsprozesse in zufälliger Umgebung, bedingt auf ¨ Uberle-ben, simuliert. Dazu wird eine Konstruktion nach [Gei99] angewendet. Diese erlaubt es, Galton-Watson Bäume in variierender Umgebung, bedingt auf Überleben, entlang einer Ahnenlinie zu konstruieren. Der Fall von Nachkommenverteilungen mit gebrochen-linearen Erzeugendenfunktionen, auf den wir uns in Kapitel 5 beschränken, erlaubt die explizite Berechnung der benötigten Verteilungen. Als Anwendung von Satz 0.3 können nun intermediär subkritische Verzeweigungsprozesse, bedingt auf Überleben, wie folgt simuliert werden: Zunächst wird die Umgebung zufällig bestimmt, und zwar als Irrfahrt, bedingt darauf ihr Minimum am Ende anzunehmen. Anschließend wird, der Geiger-Konstruktion folgend, ein Verzweigungsprozess in dieser Umgebung, bedingt auf Überleben, simuliert.

Zum Abschluss wird in einem Ausblick auf aktuelle Forschung verwiesen. Im Anhang befinden sich einige technische Resultate.

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Chapter 1

Introduction

1.1 Historical remarks

Think of a population of apomictic9plants having a one year life cycle. Each year, the weather conditions (the environment) vary, which influences the reproductive success of the plants. Given the climate, all plants reproduce according to the same mechanism. In the simplified model here, the environment is assumed to be independently and identically distributed (i.i.d.). Thus, in each generation, an offspring distribution is chosen at random, independently from one generation to the other. This is the toy model for a branching process in random environment (BPRE) (see Figure 1.1 for an example).

BPREs have first been introduced in [SW69] and [AK71]. Initially, they have mainly been studied under the assumption of i.i.d. offspring distributions which are geometric or, more generally, have generat-ing functions which are linear fractional (see [Koz76], [Afa80]). In recent years, the case of general offspring distributions has attracted attention (compare [GK00], [BGK05], [AGKV05a], [AGKV05b], [VK08], [Ban09], [Afa10], [ABKV10]), as well as special topics like large deviations, e.g. [Koz06], [BB09] and [HL10].

A list of older results is [Tan77], [Tan78], [CT84], [Gui85], [Tan88], [GZ91], [Ham92], [Afa93], [Koz95], [Liu96], [BV97], [DH97a], [Afa97], [VD97],[Afa98], [Afa01a], [VD02], [DGV04], [VD04].

In the next chapter, the mathematical model of a BPRE is described more in detail.

1.2 The model

In this section, the formal definition of the BPRE described in the preceding paragraph is presented and some basic properties of the model are described. By ∆ we denote the space of all probability measures on N0:= {0, 1, 2, 3, . . .}. Equipped with the metric of total variation10, ∆ is a Polish space. Let Q be a

random variable taking values in ∆. An infinite sequence Π = (Q1, Q2, . . .) of i.i.d. copies of Q is called

a random environment and Qn the offspring distribution in generation n − 1.

Definition. A process Z = (Z0, Z1, . . .) with values in N0 is called a branching process in random

environment Π, if Z0 is independent of Π and if, given Π, Z is a Markov chain and for every n ≥ 1,

z ∈ N0, q1, q2, . . . ∈ ∆

L Zn|Zn−1= z, Π = (q1, q2, . . .) = L(ξ1+ · · · + ξz) , (1.1)

where ξ1, ξ2, . . . are i.i.d. random variables with distribution qn.

Zn is called the nth generation size.

Fine properties of Z are mainly determined by an auxiliary process, called associated random walk, which depends on the mean offspring number in each generation.

9_{apomoxis: Reproduction without fertilization} 10_{For q}

1, q2∈ ∆, the total variation metric is defined by dT V(q1, q2) :=P∞k=0|q1({k}) − q2({k})|

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2 CHAPTER 1. INTRODUCTION

Figure 1.1: Illustration of a BPRE with three possible environmental states (taken from the presentation of Vincent Bansaye, ’Large deviations for branching processes in random environment’, 10. july 2009, ´

Ecole d’été de Probabilités, Saint-Flour).

Definition. Set Xn := log ∞ X y=0 y Qn({y}), n ≥ 1 .

The random walk S = (S0, S1, . . .) with initial state S0 = 0 and increments Xn = Sn− Sn−1, n ≥ 1 is

called associated random walk for the process (Zn)n∈N0.

Notice that the Xn are i.i.d. copies of the logarithmic mean offspring number

X = log

∞

X

y=0

y Q({y}) ,

which is assumed finite a.s. Thus, the conditioned means of Zn may be written as

E[Zn|Z0= z, Π] = z eX1· · · eXn

= z eSn _a.s. _(1.2)

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1.2. THE MODEL 3

Z

S

Figure 1.2: Illustration of a BPRE with two possible environmental states (good and bad).

In the theory of classical Galton-Watson processes, three cases are distinguished (see [AK71], p. 8) according to the mean offspring number

E[Z1|Z0= 1] > 1 supercritical case

E[Z1|Z0= 1] = 1 critical case

E[Z1|Z0= 1] < 1 subcritical case .

For BPREs, a different classification is needed. The supercritical, critical and subcritical cases are distinguished according to the drift of the associated random walk (see e.g. [BGK05]). First, if S has positive drift (i.e. limn→∞Sn = ∞ a.s., see [Fel87]), E[Zn|Π] → ∞ a.s. as n tends to infinity. This is

called the supercritical case. Second, if S has negative drift (i.e. limn→∞Sn = −∞ a.s.) the process

is called subcritical. Finally, if S is an oscillating random walk (meaning lim supn→∞Sn= ∞ a.s. and

lim infn→∞Sn= −∞ a.s.), the BPRE is called critical.

In the classical works on BPREs ([SW69], [AK71]), it has been assumed that S has finite mean. Then Z is called supercritical, subcritical or critical according as E[X] > 0, E[X] < 0 or E[X] = 0. Recently, the assumption of the existence of E[X] has been dropped (see [AGKV05b] for the strongly subcritical and [AGKV05a] for the critical case).

A nice survey of the situation in the critical and subcritical cases and an explanation of the heuristics can be found in [BGK05] which we will recall in the sequel.

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4 CHAPTER 1. INTRODUCTION although there are differences which we will explain later. In the critical and subcritical cases, the population becomes extinct with probability 1. This is an immediate consequence of a first moment estimate. For all m ≤ n,

P(Zn> 0|Π) ≤ P(Zm> 0|Π) ≤ eSm a.s. and thus P(Zn> 0|Π) ≤ min m≤ne Sm _{= exp min} m≤nSm a.s. (1.3)

For critical and subcritical BPREs, this implies P(Zn > 0|Π) → 0 a.s. and thus P(Zn → 0) = 1. In

contrast to classical Galton-Watson processes, the converse is not always true. Even in the supercritical case, it may happen that the process dies out a.s. (within only few generations) due to random fluctua-tions, a fact which will be described more in detail in the next section.

Let us shortly explain the heuristics behind the classification of BPREs. For simplicity, assume that E[X2] < ∞ and that (1.3) gives the correct order of decay of the survival probability (up to a constant), that is

P(Zn> 0) ∼ c E exp min m≤nSm.

It then remains to analyze the asymptotic of E exp minm≤nSm. In the critical case, that is for

E[X] = 0, there will only be a considerable contribution to the expectation if minm≤nSmis close to zero.

In the finite variance case, it is well-known that the probability of {minm≤nSm≥ 0} is of the order n−1/2

(see e.g. [GKV03]) and we expect

P(Zn> 0) ∼ c P min

1≤m≤nSm≥ 0

∼ c0 n−1/2 .

for some constant c0 > 0. As it will be detailed in the next section, this result is also true in a more general context and does not require finite variance of the associated random walk.

Introduce

Mn := max

1≤j≤nSj

Ln := min

1≤j≤nSj .

By τn, we denote the first time, when the minimum of S0, . . . , Sn is attained

τn := min{0 ≤ k ≤ n|Sk = min{S0, . . . , Sn}} n ≥ 0 . (1.4)

Now consider a subcritical BPRE where the associated random walk has negative drift. Then the prob-ability of {Ln ≥ 0} is exponentially small and it is more complicated to estimate the asymptotic of

E exp Ln. For this, we will use a change of measure and define

β := sup0 ≤ s < ∞

EXesX = 0 . (1.5)

For simplicity, assume that β is finite and that the supremum is attained. Then E[XeβX] = 0 .

As it turns out, there are three different regimes, depending on β. Namely, we distinguish the weakly subcritical (0 < β < 1), the intermediately subcritical (β = 1), and the strongly subcritical case (β > 1). They are characterized by different asymptotics of P(Zn > 0) and different limit behavior,

conditioned on survival. New limit theorems for the intermediately subcritical case will be proved in Chapter 3.

In the following, a major tool will be the change of measure. If we make an exponential change of measure with parameter β, we get a new measure under which S does not have any drift. More precisely, let ϕ : Rn+1_{× ∆}n _{→ R be a bounded and measurable function. Then the probability measure P is}

defined by

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1.2. THE MODEL 5 associated random walk P(Zn> 0) Classification

E[X] > 0 P(Zn> 0, ∀n ≥ 0) > 011 supercritical

E[X] = 0 P(Zn> 0) ∼ θ1P(Ln≥ 0) critical

E[X] < 0 P(Zn> 0) ∼ e−γn+o(n) subcritical

Table 1.1: Classification of BPREs. The constant 0 < θ1 < ∞ depends on properties of the offspring

distributions, and 0 < γ = −_n1_{log E[e}Ln_{] (see Lemma 4.3.5 in Chapter 4).}

associated random walk P(Zn> 0) ∼ Classification

E[XeX] > 0 θ2P(Ln≥ 0) weakly subcr.

E[XeX] = 0 θ3e−γnP(τn= n) intermediately subcr.

E[XeX] < 0 θ4E[eX]n strongly subcr.

Table 1.2: Classification of subcritical BPREs (E[X] < 0). The positive constants θ2, θ3 and θ4 depend

on properties of the offspring distributions, and 0 < γ = −_n1_{log E[e}Ln_{] (see Lemma 4.3.5 in Chapter 4).}

with

γ := − log Ee(β∧1)X_. _(1.7)

In the weakly and intermediately subcritical cases, (1.5) translates to E[X] = 0 and S becomes a recurrent random walk under P. The change of measure then allows us to use many important properties of recurrent random walks. In the strongly subcritical case (β > 1), E[X] < 0, thus S has also a negative drift with respect to P. There, it is not suitable to change to a measure under which S is recurrent. This will become clear later. We only give the heuristics here (see [BGK05]) and assume that (1.3) gives the right order of (up to a constant) decay of the survival probability. Then the change of measure yields (with some positive constant c)

P(Zn > 0) ∼ c e−γnE[eLn−(β∧1)Sn] .

There will only be a considerable contribution to the expectation if Ln− (β ∧ 1)Sn is close to zero. Now

there are three different cases:

• 0 < β < 1. Then, Ln− βSn will only be small if both Ln and Sn are close to zero. The probability

of having such an excursion of length n is (up to a constant factor) asymptotically equal to n−3/2 for a zero mean and finite variance random walk (see e.g. [GKV03]). Thus here, one expects

E[eLn−βSn_] _≈ _n−3/2 _.

• β = 1. Here, Ln−βSnwill be small if Lnand Snare close to each other which essentially is the case if

the random walk has its minimum close to the end, {τn = n}. By duality, P(τn= n) = P(Mn< 0).

It is known for zero mean and finite varinance random walks that the latter probability is of the order n−1/2 (see e.g. [GKV03]). Thus

E[eLn−Sn_] _≈ _n−1/2 _.

• β > 1. In this case, E[X] < 0. Since for a random walk with negative drift, Ln− Sn will be of

constant order, one expects

E[eLn−Sn_] _≈ _const.

In the strongly subcritical case, the probability of survival decreases with the same order as the ex-pected generation size, E[Zn|Z0 = 1] = E[eSn] = E[eX]n. This resembles the behavior of subcritical

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6 CHAPTER 1. INTRODUCTION Galton-Watson processes. However, BPREs are not classified according to the expected generation size E[Zn] = E[eX]n. If E[eX] > 1, the process may be supercritical, critical or weakly subcritical, depending

on the distribution of X. E[eX_{] < 0 only implies that the BPRE is subcritical.}

Table 1.1 and 1.2 summarize the classification (for simplicity assume that the offspring distributions have finite variance and that E[|X|] < ∞ and E[|X|eX_{] < ∞; the results hold true under more general}

conditions, but we refrain from giving details here).

The different regimes of BPREs will be characterized in the next chapter by recalling some known results. The intermediately subcritical case is studied in Chapter 3. There, new limit theorems will be proved which describe properties of an intermediately subcritical BPRE, conditioned on survival. In Chapter 4, large deviations of BPREs are analyzed, both for offspring distributions with geometrically bounded tails as well as for heavy-tailed offspring distributions. In Chapter 4.4, a short outlook on the problem of analyzing lower deviations of supercritical BPREs is provided. In Chapter 4.5, upper large deviations of Z, conditioned on the environment (the so-called quenched approach), are studied. Finally, in Chapter 5 a simulation algorithm for conditioned BPREs is explained and in Chapter 6, a short outlook on current research is presented. Some technical results are proved in the appendix.

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Chapter 2

Classification and known results for

BPREs

In the following, for sequences (dn) and (mn), we use the notation dn ∼ mn if dn/mn → 1 as n → ∞.

All limit theorems presented in this chapter are under the law P (i.e. averaged over the environment) which is what is called the annealed approach. In contrast to this, under the quenched approach, limit theorems conditioned on the environment Π are developed.

2.1 The supercritical case

Suppose that E[|X|] < ∞ and E[X] > 0, that is S has a positive drift. In the case of classical Galton-Watson processes, supercritical processes have a positive survival probability. For BPREs, the random fluctuations of the environment can cause a.s. extinction of the process even if E[X] > 0. For the survival of the process, an additional integrability condition for the probability that an individual has no offspring, Q({0}), is needed. The following theorem is proved in [SW69] and [Smi68].

Theorem 2.1.1. (Smith and Wilkinson (1968/69)) Suppose E[|X|] < ∞. Then the BPRE (Zn)n∈N0

has a positive survival probability,

lim n→∞P(Zn> 0) > 0 , iff E[X] > 0 and E h log 1 − Q({0})i > −∞ . (2.1)

The second condition in (2.1) assures that catastrophic events, meaning the probability of an individual having no offspring is very close to one, are sufficiently improbable. Heuristically, if the second condition in (2.1) is not met, the process will die out a.s. due to such ’catastrophes’ within a few generations. For a simple example when (2.1) is not fulfilled, consider the set of distributions A ⊂ ∆ with just one free parameter, only putting mass onto two points: for all q ∈ A, l := 1 − q(0) ∈ (0, 1) and q(d2/le) = l. Thus the mean of this distribution is

mq = d2/le · l > 1 .

Now the parameter is chosen at random according to the following distribution with parameter α ∈ (0, 1), P(L ≤ x) = c (− log(x))−α ,

where c > 0 is the norming constant. Then E[X] = E log d2/Le · L > 0 and E

h

log 1 − Q({0})i

= E[log L] = −∞ . 7

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8 CHAPTER 2. CLASSIFICATION AND KNOWN RESULTS FOR BPRES

2.2 The critical case

The critical case is treated in several papers (see [Koz76], [Afa93], [Afa97], [GK00], [AGKV05a], [BDKV10]). The following simulation with R12_{gives a first impression of the situation in the critical case. Survival of}

the process is essentially realized by a ‘good’ environment (i.e. {Ln≥ 0}). Conditioned on survival until

generation n, S behaves like a random walk conditioned on staying positive and Z exhibits supercritical growth. This will be described more in detail by the theorems presented in this section.

50 100 150 200 250 300 0 10 20 30 40

Example of a BPRE: Generation size Z

lo g10

((

Z

))

n

Figure 2.1: Example: A critical BPRE with geometric offspring distributions.

Here, we recall some results from [AGKV05a] on the asymptotic of the survival probability and the asymptotic behavior of the process, conditionend on survival. The following theorems are proved under the assumption that the associated random walk fulfills Spitzer’s condition, that is

Assumption 2.1. There exists a number 0 < ρ < 1 such that 1 n n X m=1 P(Sm> 0) → ρ as n → ∞ .

It says that the expected proportion of time that S spends within the positive real half line up to time n, converges as n → ∞ to some value in (0, 1). Any random walk fulfilling Assumption 2.1 is of the oscillating type.

Next we define renewal function u(x) :=

1 +P∞

k=1P(−Sk ≤ x, Mk < 0) , if x ≥ 0

0 , else . (2.2)

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2.2. THE CRITICAL CASE 9 50 100 150 200 250 300 0 10 20 30

Example of a BPRE: Associated random walk S

S

n

Figure 2.2: Example: Associated random walk S for the BPRE displayed in Figure 2.1.

The function u will be explained in detail later. Here just note that for any oscillating random walk,

E[u(x + X)] = u(x) , x ≥ 0 . (2.3)

Furthermore, one needs some regularity of the offspring distributions Q. For this, we introduce the standardized truncated second moment of Q,

ζ(a) := ∞ X y=a y2Q({y})/m(Q)2 , _{a ∈ N}0 , (2.4) where m(Q) := ∞ X k=0 k Q({k}) .

The second assumption now assures some regularity of the offspring distributions. Assumption 2.2. For some > 0 and some a ∈ N0, we assume

E h

log+ζ(a)1/ρ+i

< ∞ and _Ehu(X) log+ζ(a)1+i

< ∞ , (2.5)

where log+x := log(max(x, 1)).

Some examples where Assumption 2.2 is fulfilled are:

• Q has uniformly bounded support, i.e. there exists a c < ∞ such that Q({0, 1, . . . , c}) = 1 P–a.s. In particular, Assumption 2.2 is trivially fulfilled for any binary branching process in random envi-ronment, that is when an individual has either two children ore none.

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10 CHAPTER 2. CLASSIFICATION AND KNOWN RESULTS FOR BPRES • By (2.3), E[u(X)] = u(0) < ∞, thus Assumption 2.2 is fulfilled if ζ(a) is a.s. bounded from above.

In case of Poisson or geometric distributions, the standardized second factorial moment, η :=

∞

X

y=0

y(y − 1)Q({y})/m(Q)2 ,

is a constant (η = 1 for a Poisson distribution, η = 2 for a geometric distribution). As ζ(2)/2 ≤ η ,

Assumption 2.2 is fulfilled if Q is a.s. a Poisson or a geometric distribution.

• It is possible to get rid of the renewal function u in Assumption 2.2. It is known that u(x) = O(x) as x → ∞ (see [Fel87], Chapter XII) and by definition, u(x) = 0 for x < 0. Thus, using H¨older’s inequality, Assumption 2.2 is fulfilled if

E[(X+)p] < ∞ and E(log+ζ(a))q < ∞ for some p > 1 and q > max{ρ−1, p/(p − 1)}.

Alternatively, if one has more regularity of the tails of X, one can replace the assumptions by the following two conditions.

Assumption 2.3. The distribution of X has finite variance or (more generally) belongs to the domain of attraction of some stable law s(·) with index α ∈ (0, 2]. The limit law is not a one-sided stable law, that is, 0 < s(R+_{) < 1.}

This means that there is an increasing sequence of positive numbers an = n1/αln

with a slowly varying sequence l1, l2, . . . such that for n → ∞

P(Sn/an∈ dx) → s(x)dx .

Remark. In general, Assumption 2.1 is less restrictive than Assumption 2.3. However, if X− has finite second moment, Assumption 2.3 is equivalent to Spitzer’s condition (cf. [Don77]).

By this gain of regularity, Assumption 2.2 may be relaxed to Assumption 2.4. For some > 0 and some a ∈ N0, let

E h

log+ζ(a)α+i

< ∞ , where log+x := log(max(x, 1)).

The following theorems have been proved in [AGKV05a]. The first one describes the assymptotic of the nonextinction probability.

Theorem 2.2.1. (Afanasyev, Geiger, Kersting and Vatutin (2005)) Assume Assumptions 2.1 and 2.2 or 2.3 and 2.4. Then there exists a number 0 < θ < ∞ such that

P(Zn > 0) ∼ θ P(min{S1, . . . , Sn} ≥ 0) as n → ∞ .

The asymptotic of the nonextinction probability is again – up to the constant factor θ – completely determined by properties of the associated random walk. The theorem essentially says that survival until time n is -up to the constant factor- as improbable as the minimum of the associated random walk being nonnegative. This reflects the fact that by (1.3), the nonextinction probability is small if S has a low minimum. Conditioning on survival, that is on the event {Zn > 0}, is essentially the same as conditioning

on {min{S1, . . . , Sn} ≥ 0}. A more detailed description of this phenomenon is provided by Theorems

2.2.4 and 2.2.5.

Under Spitzer’s condition (Assumption 2.1), the asymptotic behavior of the minimum of a random walk is well-known, which leads to the following corollary.

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2.2. THE CRITICAL CASE 11 Corollary 2.2.2. (Afanasyev, Geiger, Kersting and Vatutin (2005)) Assume Assumptions 2.1 and 2.2 or 2.3 and 2.4. Then there is a slowly varying sequence l1, l2, . . . such that

P(Zn > 0) ∼ θ ln n−(1−ρ) , as n → ∞ .

Let us now look at the behavior of the branching process, conditioned on survival. As it turns out, conditioned on {Zn > 0}, the generation size process Z0, Z1, Z2, . . . shows a kind of ‘supercritical’

be-havior. For classical Galton-Watson processes, this means that Zn/eSn converges a.s. to some typically

nondegenerated, positive random variable. In our case, the conditional distribution of the environment Π, given {Zn> 0}, changes with n.

Thus, the following theorem is formulated for the rescaled generation size process Xr,n _{= (X}r,n t )0≤t≤1,

with r ≤ n, r, n ∈ N0. It is defined by

X_tr,n := Zr+b(n−r)tc

eSr+b(n−r)tc , 0 ≤ t ≤ 1 . (2.6)

Theorem 2.2.3. (Afanasyev, Geiger, Kersting and Vatutin (2005)) Assume Assumptions 2.1 and 2.2 or 2.3 and 2.4. Let r1, r2, . . . be a sequence of natural numbers such that rn ≤ n and rn → ∞.

Then

L(Xrn,n_|Z

n> 0) d

→ L((Wt)0≤t≤1) as n → ∞ ,

where the limiting process is a stochastic process with a.s. constant paths, that is, P(Wt= W f or all t ∈ [0, 1]) = 1 for some random variable W . Furthermore,

P(0 < W < ∞) = 1 .

By →, we denote the weak convergence with respect to the Skorohod topology in the space D[0, 1] ofd c`adl`ag functions13 _{on the unit interval. By Theorem 2.2.3, again the growth of Z is mainly determined}

by the associated random walk, that is by the sequence (eSn₎

n>0. The process thus exhibits supercritical

behavior. The fine structure of the environment only affects the random variable W .

As mentioned above, conditioning on {Zn > 0} affects the environment and thus changes the behavior

of S. The following two theorems describe this phenomenon. Recall that by τn, we denote the first time,

when the minimum of S0, . . . , Sn is attained (see Definition (1.4)),

τn := mini ≤ n|Si= min{S0, . . . , Sn} , n ≥ 0 .

Theorem 2.2.4. (Afanasyev, Geiger, Kersting and Vatutin (2005)) Assume Assumptions 2.1 and 2.2 or 2.3 and 2.4. Then, as n → ∞,

L (τn, min{S0, . . . , Sn})|Zn > 0

converges weakly to some probability measure on N0× R−0.

This theorem states that, conditioned on survival, the associated random walk has its (global) minimum at some finite time.

A more detailed description is proved in the situation of Assumption 2.3.

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12 CHAPTER 2. CLASSIFICATION AND KNOWN RESULTS FOR BPRES 50000 100000 150000 200000 0 200 400 600 800

Example: Simple random walk

S

n

Figure 2.3: Example: A simple random walk conditioned to stay nonnegative.

Theorem 2.2.5. (Afanasyev, Geiger, Kersting and Vatutin (2005)) Assume Assumptions 2.3 and 2.4. Then there exists a slowly varying sequence l1, l2, . . . such that

L((n−1/αlnSbntc)0≤t≤1|Zn> 0) d

→ L(L+₎ _{as n → ∞ ,}

where L+ denotes the meander of a strictly stable L´evy process L with index α.

Let us shortly explain the process L+_{. Convergence of a conditioned Brownian motion to the Brownian}

meander has been proved in [DIM77] (see also e.g. [Don85] and [Dur78] for convergence of conditioned Markov chains). Essentially, L+ _{is a strictly stable L´}_{evy process, conditioned to stay positive on the}

time intervall (0, 1]. L+ _{is the limiting process of {(n}−1/α_l

nSbntc)0≤t≤1|Ln ≥ 0}. The existence and

characterization of this limit can be found in [Don85]. Figure 2.3 illustrates a random walk conditioned to stay nonnegative. In case of finite variance, α = 2, L+_{is the meander of a standard Brownian motion.}

To sum up, a critical BPRE behaves, conditioned on survival, similar to a supercritical branching process. Conditioned on survival, the environment is ‘good’, i.e. the rescaled associated random walk converges to L´evy process conditioned to stay nonnegative.

2.3 The subcritical cases

2.3.1 The strongly subcritical case

This case has been discussed intensively in [AGKV05a]. Here we recall the main results. The existence of β (defined in 1.5) is actually not needed. It suffices that the following condition is fulfilled:

Assumption 2.5.

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2.3. THE SUBCRITICAL CASES 13 5 10 15 0 5 10 15 20

Example of a BPRE: Generation size Z

Z

n

Figure 2.4: Example: A strongly subcritical BPRE with geometric offspring distributions See Figures 2.4 and 2.5 for sample simulations with R. Additionally, an integrability condition for the offspring distributions is required, namely:

Assumption 2.6.

E[Z1log+Z1] < ∞ ,

where log+x := log(max(x, 1)).

Note that Assumption 2.6 implies E[Z1] = E[m(Q)] < 1 and E[log Z1] = E[log m(Q)] < 0. An

assump-tion on the standardized second factorial moment of Z1, conditioned on Π (see Definition (2.4)) with

Assumption 2.5 already implies Assumption 2.6. Let for some a > 0,

E[m(Q) log+ζ(a)] < ∞ . (2.7)

Then by Jensen’s inequality and the definition of ζ (see (2.4)),

∞ X y=1 log yyQ({y}) m(Q) ≤ a log a + log ∞ X k=a y2Q({y}) m(Q)

≤ a log a + log+m(Q) + log+ζ(a) P − a.s. Multiplying both sides with m(Q) and taking the expectation yields

E[Z1log+Z1] ≤ a log aE[m(Q)] + E[m(Q) log+m(Q)] + E[m(Q) log+ζ(a)] < ∞ .

For examples where (2.7) is fulfilled, we refer to Section 2.2. In particular, (2.7) is abundant if Q is P–a.s. a geometric or a Poisson distribution. Also note that taking P(Q = q) = 1 for some q ∈ ∆ yields a

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14 CHAPTER 2. CLASSIFICATION AND KNOWN RESULTS FOR BPRES 5 10 15 −10 −8 −6 −4 −2 0

Example of a BPRE: Associated random walk S

S

n

Figure 2.5: Example: Associated random walk S.

classical Galton-Watson process. If m(q) < 1, Assumption 2.5 is fulfilled. The second assumption is then well-known (see e.g. [AN72, p. 45]) to be a sufficient condition for

P(Zn> 0) ∼ c E[Zn] = c m(q)n .

Like in the case of classical Galton-Watson processes, the first moment estimate already gives the right decay rate of the survival probability in the strongly subcritical regime (see [GL01], originally proved in [DH97b] under an additional moment condition).

Theorem 2.3.1. (Guivarc’h and Liu (2001)) Under Assumptions 2.5 and 2.6, there is a 0 < θ ≤ 1 such that

P(Zn > 0) ∼ θ E[Zn] as n → ∞ .

The next theorem, proved in [GKV03], states that – as in the case of subcritical Galton-Watson processes – the generation size has a weak limit, conditioned on nonextinction.

Theorem 2.3.2. (Geiger, Kersting and Vatutin (2003)) Under Assumptions 2.5 and 2.6, there is a probability measure υ with weights υz, such that

lim

n→∞P(Zn= z|Zn> 0) = υz, z ∈ N .

The previous theorem, together with Fatou’s lemma, yields m(υ) ≤ θ−1 < ∞ so that the expectation with respect to υ is finite. In fact, as it is proved in [AGKV05a], m(υ) = θ−1. The next theorems describe the behavior of a strongly subcritical branching process in random environment, conditioned on survival, more in detail. The following theorem says that, conditioned on nonextinction, the offspring

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2.3. THE SUBCRITICAL CASES 15 distributions are independent in the limit with respect to P (recall definition (1.6)). Moreover, excursions of the associated random walk vanish in the skaling limit. This means that the process does not exhibit any supercritical behavior, conditioned on survival.

Theorem 2.3.3. (Afanasyev, Geiger, Kersting and Vatutin (2005)) Assume 2.5 and 2.6, let in,j, n ∈ N, 1 ≤ j ≤ k be nonnegative integers with 1 ≤ in,1< in,2< · · · < in,k≤ n, and n − in,k→ ∞ as

n → ∞. Then for every k ∈ N and Borel sets B1, . . . , Bk ⊂ ∆,

lim n→∞P(Qin,1 ∈ B1, . . . , Qin,k ∈ Bk|Zn> 0) = k Y j=1 P(Q ∈ Bj) . Moreover, L (n−1S_bntc)0≤t≤1 Zn > 0 d → L (tE[X])0≤t≤1

with respect to the Skorohod topology.

The next result characterizes the dynamics of the generation size process Z, conditioned on nonextinction. No scaling of Z is necessary, which essentially means that, conditioned on {Zn> 0}, the population stays

small throughout the time intervall from 0 to n. In the special case when Q has linear fractional generating functions, the following theorem has first been obtained in [Afa01b]. A more general version of it has been established in [AGKV05a].

Theorem 2.3.4. (Afanasyev, Geiger, Kersting and Vatutin (2005)) Under Assumptions 2.5 and 2.6 and for any 0 < t1< · · · < tk< 1, as n → ∞

L (Zbnt1c, . . . , Zbntkc)

Zn> 0 d

→ (W1, . . . , Wk) ,

where W1, W2, . . . are i.i.d. copies of some random variable W with

P(1 ≤ W < ∞) = 1 .

Summing up, survival of the process in the strongly subcritical regime is typically not realized by a favorable environment (i.e. min{S1, . . . , Sn} ≥ 0 or bounded), although the environment is more favorable

than it is typically for the unconditioned process (i.e. E[X] > E[X]). In the limit and conditioned on nonextinction, the associated random walk behaves in a completely deterministic manner and decays linearly according to its expectation with respect to the measure P and the offspring distributions at different times become independent. Survival is realized by exceptional offspring numbers within this environment.

2.3.2 The weakly subcritical case

The results in this section are based on joint work with Valery Afanasyev14, G¨otz Kersting and Vladimir Vatutin14and published in [ABKV10]. As it turns out, methods developed in [AGKV05a] for criticality, can also be used for weak subcriticality. Conditioned on survival, the process exhibits ‘supercritical’ behavior (see Figures 2.6 and 2.7 for simulations).

Let us briefly state the results from [ABKV10]:

Assumption 2.7. The process Z is weakly subcritical, that is there is a number 0 < β < 1 such that E[XeβX] = 0 .

As explained in Section 1.2, this allows to change to a measure P according to (1.6) and S becomes a recurrent random walk under P.

As to the regularity of the distribution of X the following assumptions are needed.

Assumption 2.8. The distribution of X has finite variance with respect to P or (more generally) belongs to the domain of attraction of some stable law s(·) with index α ∈ (1, 2]. It is non-lattice.

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16 CHAPTER 2. CLASSIFICATION AND KNOWN RESULTS FOR BPRES 50 100 150 200 250 0 5 10 15

Example of a BPRE: Generation size Z

lo g10

((

Z

))

n

Figure 2.6: Example: Logarithm of the population size of a weakly subcritical BPRE with geometric offspring distributions.

The last assumption on the environment concerns the standardized truncated second moment of Q. Recall definition (2.4), ζ(a) := ∞ X y=a y2Q({y}).m(Q)2 , _{a ∈ N .}

Assumption 2.9. For some ε > 0 and some a ∈ N

E[(log+ζ(a))α+ε] < ∞ , where log+x := log(max(x, 1)).

Compare Section 2.2 for examples where this assumption is fulfilled.

Now the main results for the weakly subcritical case will be described briefly. The first theorem describes the asymptotic behavior of the nonextinction probability at generation n.

Theorem 2.3.5. (Afanasyev, B., Kersting and Vatutin (2009)) Under Assumptions 2.7 to 2.9, there exists a number 0 < κ < ∞ such that

P(Zn > 0) ∼ κ P(min(S1, . . . , Sn) ≥ 0) as n → ∞ .

Recall that, as it has been mentioned in Section 2.2, under Assumption 2.8, there exists an increasing sequence of positive numbers

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2.3. THE SUBCRITICAL CASES 17 50 100 150 200 250 0 2 4 6 8 10 12 14

Example of a BPRE: Associated random walk S

S

n

Figure 2.7: Example: Associated random walk S.

with a slowly varying sequence l1, l2, . . . such that for n → ∞,

P(Sn/an ∈ dx) → s(x)dx .

Then from Theorem 2.3.5, the following corollary results:

Corollary 2.3.6. Under Assumptions 2.7 to 2.9, there is a number 0 < κ0< ∞ such that

P(Zn> 0) ∼ κ0

e−γn n an

, where γ = − log E[eβX_].

The next theorem yields convergence of the laws of (Zn)n∈N, conditioned on survival.

Theorem 2.3.7. (Afanasyev, B., Kersting and Vatutin (2009)) Under Assumptions 2.7 to 2.9, the conditional laws L(Zn| Zn > 0), n ≥ 1, converge weakly to some probability distribution on the natural

numbers. Moreover the sequence E[Zϑ

n| Zn > 0] is bounded for any ϑ < β, implying convergence to the

corresponding moment of the limit distribution.

The last theorem describes the limiting behavior of the rescaled generation size process e−Sk_Z

k for

rn ≤ k ≤ n − rn, where (rn) is a sequence of natural numbers with rn → ∞ (and rn < n/2). Thus we

consider the process Yn= {Y_tn, t ∈ [0, 1]}, defined by

Y_tn:= exp(−Srn+b(n−2rn)tc)Zrn+b(n−2rn)tc .

This process has asymptotic paths of a constant random value. More precisely, the following statement holds:

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18 CHAPTER 2. CLASSIFICATION AND KNOWN RESULTS FOR BPRES Theorem 2.3.8. (Afanasyev, B., Kersting and Vatutin (2009)) Under Assumptions 2.7 to 2.9, there is a process {Wt, t ∈ [0, 1]} such that as n → ∞

L Yn t , t ∈ [0, 1] Zn> 0 d → L Wt, t ∈ [0, 1]

weakly in the Skorohod space D[0, 1]. Moreover, there is a random variable W such that Wt= W a.s. for

all t ∈ [0, 1] and

P{0 < W < ∞} = 1 . Weaker versions of these results can be found in [Afa98] and [GKV03].

Thus we have the following scenario in the weakly subcritical case: Conditioned on {Zn > 0}, Z starts

growing in a favorable environment at the beginning and roughly up to time bnc Z exhibits supercritical growth. It then follows the value of eSk_{= E[Z}

k| Π] in a completely deterministic manner, up to a random

factor W > 0. It is due to the fact that Sk takes large values there. This behavior persists as long as S

takes large values. Close to n, in the manner of an excursion (see [DIM77]) S returns to zero and the environment becomes more and more unfavorable. The values of Z are decreasing so that Z is again small close to generation n.

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Chapter 3

The intermediately subcritical case

3.1 Introduction and main results

Here we study the intermediately subcritical case (see Figures 3.1 and 3.2 for simulations). As it already can be seen from the sample simulation, in contrast to the other cases, the population stays small throughout most of the time, but still, there are supercritical periods when the population grows very large. In this chapter, this ‘alternating’ behavior is studied.

200 400 600 800 1000 0 2 4 6 8

Branching Process Z

lo g10

((

Z

))

n

Figure 3.1: Example: An intermediately BPRE Z conditioned on {Z1000> 0} (based on the simulation

scheme described in Chapter 5). Assumption 3.1.

E[XeX] = 0 . (3.1)

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20 CHAPTER 3. THE INTERMEDIATELY SUBCRITICAL CASE 200 400 600 800 1000 −40 −30 −20 −10 0 10

Associated random walk

S

n

Figure 3.2: Example: Associated random walk S for the process in Figure 3.1 (simple random conditioned on having its minumum at the end).

From (3.1) results E[X] < 0 (except for the degenerated case X = 0 a.s.). By Jensen’s inequality it follows that (see [AGKV05b])

E[XeX] = E log(eX) · eX ≥ log

E[eX]

· E[eX] and thus E[eX_{] < 1.}

As explained in Chapter 1, (3.1) suggests to change to the measure P with expectation E defined by E [Φ(Q1, . . . , Qn, Z0, . . . , Zn)] = γ−nEΦ(Q1, . . . , Qn, Z0, . . . , Zn)eSn−S0

with

γ = EeX < 1.

E[X] = 0 follows from (3.1). Thus S becomes a recurrent random walk under P. We need a regularity assumption for the distribution of X.

Assumption 3.2. The distribution of X has, with respect to P, finite variance or (more generally) belongs to the domain of attraction of some strictly stable law with index α ∈ (1, 2]. For convenience, it is non-lattice.

We recall a few consequences of Assumption 3.2 (see e.g. [ABKV10]). There is an increasing sequence of positive numbers

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3.1. INTRODUCTION AND MAIN RESULTS 21 with a slowly varying sequence l1, l2, . . . such that

P(Sn/an∈ dx) → s(x)dx

weakly, where s(x) denotes the density of the limiting stable law. Furthermore, there is a ρ ∈ (0, 1), ρ ≤ α−1 _{such that} 1 n n X m=1 P(Sm> 0) → ρ as n → ∞ .

(see e.g. [AGKV05a]). The next assumption concerns the standardized truncated second moment of Q. Recall from (2.4) that

ζ(a) =

∞

X

y=a

y2Q({y}).m(Q)2, _{a ∈ N ,}

where mQ =P∞y=0ymQ({y}).

Assumption 3.3. There are constants 0 < < ∞ and a ∈ N such that E(log+

ζ(a))α+ < ∞ . For examples where this assumption is met see Chapter 2.2.

Recall that τn is the time of the first minimum of (S0, . . . , Sn) (see Definition (1.4) in Chapter 1),

τn := min0 ≤ k ≤ n|Sk = min{S0, . . . , Sn}

, n ≥ 0 . The following theorem is proved in [Vat04] under somewhat stronger conditions. Theorem 3.1.1. Under Assumptions 3.1 to 3.3,

P(Zn> 0) ∼ γnθ P(τn= n)

for some 0 < θ < ∞.

Remark. There is an expression of θ in terms of a sum of expectations with respect to E (see proof of Theorem 3.1.1, Section 3.4).

Our next theorem describes the number of times – conditioned on survival – when there is only one individual left. We assume that, with positive probability, there are distributions allowing individuals to have no child or exactly one child.

Assumption 3.4.

EQ({1})Q({0}) > 0 .

Then the number of times when there is just one individual left is of the same order as the number of strictly descending ladder points of an oscillating random walk conditioned on {τn= n}.

Theorem 3.1.2. Under Assumptions 3.1 to 3.4, there is a slowly varying sequence b1, b2, . . . such that

E h ]{k|Zk= 1} Zn > 0 i = Θ(bn n1−ρ) .

By xn = Θ(yn) we denote that the sequence xn is of the same order as yn as n → ∞. More precisely,

there are constants c1, c2∈ R+ such that

c1 ≤ lim inf n→∞ xn yn ≤ lim sup n→∞ xn yn ≤ c2 .

As it is already indicated by Theorem 3.1.1, conditioned on nonextinction, the associated random walk converges to a L´evy process conditioned on having its minimum at the end.

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22 CHAPTER 3. THE INTERMEDIATELY SUBCRITICAL CASE Theorem 3.1.3. Under Assumptions 3.1, 3.2, and 3.3, as n → ∞,

L S_bntc/a_bntc 0≤t≤1 Zn > 0 _d → L(L†)

in distribution with respect to the Skorohod metric, where L† denotes a L´evy process conditioned to have its minimum at the end.

Essentially, this theorem says that for the associated random walk, conditioning on survival of the process Z is the same as conditioning on having the minumum at the end (see Figure 3.3). Thus, survival of the process is typically not realized by a ‘favorable’ environment, but by exceptional offspring numbers with an environment characterized by a recurrent associated random walk having its minimum close to the end.

Typically, a recurrent random walk conditioned on having its minimum close to the end may have

50000 100000 150000 −500 −400 −300 −200 −100 0

Example: Simple random walk

S

n

Figure 3.3: Example: A simple random walk conditioned to have its minimum at the end.

some long excursions between the strictly descending ladder points. During a long excursion, we expect that the process may grow very large and exhibits a behavior similiar to a weakly subcritical BPRE, conditioned on survival (see Section 2.3.2). For simulations of the conditioned process see Chapter 5. A theorem describing this limiting behavior of the rescaled generation size process has been proved in [Afa01b] in the case of geometric offspring distributions. We claim that the statement also holds in the more general case treated here, but the proof remains an open problem.

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3.2. CONDITIONAL LIMIT LAWS FOR OSCILLATING RANDOM WALKS 23 Open Problem. Let L = (L†_t)0≤t≤1 be a strictly stable L´evy process, conditioned to have its minimum

at time 1. By e1, e2, . . . we denote its excursion intervals. Let j(t) = i if t ∈ ei. Then for 0 < t1< t2<

. . . < tk< 1, as n → ∞ L Zbnt1c exp(S_bnt₁_c− mink≤bnt1cSk) , . . . , Zbntkc exp(S_bnt_k_c− mink≤bntkcSk) Zn > 0 ! d −→ L (Wj(t1), . . . , Wj(tk)) ,

where W1, W2, . . . are i.i.d. copies of some strictly positive random variable W .

3.2 Conditional limit laws for oscillating random walks

3.2.1 A change of measure

In this section, we introduce the probability measures P+_{and P}− _{which will be used in the proof of the}

limit laws (see Sections 3.3 and 3.4). Here, we state general results for oscillating random walks without referring to BPREs. Let

Mn := max

1≤j≤nSj

Ln := min

1≤j≤nSj .

Let us recall some results from fluctuation theory of random walks. Two standard references are [Spi64] and [Fel87, chaper XII]. We introduce the right-continuous functions u : R+0 → R and v : R−0 → R defined

by u(x) := 1 + ∞ X k=1 P(−Sk ≤ x, Mk < 0), x ≥ 0 , v(x) := 1 + ∞ X k=1 P(−Sk > x, Lk≥ 0), x ≤ 0 . (3.2)

In particular u(0) = v(0) = 1. Also it is known that u(x) = O(x), v(x) = O(x). Another representation of u and v uses the strictly descending ladder times, 0 =: γ

0< γ1< · · · and the weakly ascending ladder

times, 0 =: γ₀< γ₁· · · , defined by γ

i := minn > γi−1 : Sn< Sγi−1 ,

γ_i := minn > γi−1 : Sn≥ Sγ_i−1 , i ≥ 1 .

Then u(x) = 1 + ∞ X k=1 P(Sγ k≥ −x), x ≥ 0 , v(x) = 1 + ∞ X k=1 P(Sγ_k< −x), x ≤ 0 . (3.3)

Thus u(x) is the expected number of strictly descending ladder epochs that do not fall below the level −x (see Figure 3.4 for an example).

By the duality lemma, (3.2) and (3.3) are equivalent (see [Fel87, pp. 394/395]). Let us briefly recall the duality argument here for the function v as duality will also be an important tool later. In the sequel, ˆS shall denote the dual random walk, defined by

ˆ

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24 CHAPTER 3. THE INTERMEDIATELY SUBCRITICAL CASE 0 10 20 30 40 −3 −2 −1 0 1 2

Random walk with increments uniformly distributed on [−1,1]

S

n

● γγ1 ● γγ2 ● γγ3 ● γγ4 ● γγ5 ● γγ6 ● γγ7 ● γγ8● γγ9

Figure 3.4: Example: Ladder points γi above level -3 for a random walk with increments that are

uniformly distributed on [−1, 1].

We refrain from indicating the dependence on n in our notation. Now by duality, Snis a weakly ascending

ladder height if

{Sn ≥ S0, . . . , Sn≥ Sn−1} = { ˆSn≥ 0, . . . , ˆS1≥ 0} = { ˆLn≥ 0} ,

which proves the equivalence of (3.2) and (3.3).

For any oscillating random walk, both u and v are harmonic functions (see [BD94] and [AGKV05a]), that is

Eu(x + X); X + x ≥ 0 = u(x) , x ≥ 0 ,

Ev(x + X); X + x < 0 = v(x) , x ≤ 0 . (3.5)

Thus u and v can be used to construct new probability measures P+ _{and P}−_{. The construction of these}

measures is described in detail in [AGKV05a]. For the sake of completeness, we recall the construction for P−. In the sequel, we denote by Pxand Exthat the random walk starts in S0= x.

For this, assume that S is adapted to some filtration (Fn)n≥0such that Xn+1is independent of Fnfor all

n ≥ 1. Let R0, R1, . . . be a sequence of random variables with values in some state space S, also adapted

to F . The sequence

v(S0) , v(S1)1l{M1<0} , v(S2)1l{M2<0} , . . .

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3.2. CONDITIONAL LIMIT LAWS FOR OSCILLATING RANDOM WALKS 25 and measurable function g : Sn+1

→ R,

Ex[g(R0, . . . , Rn)v(Sn+1); Mn+1< 0]

= Ex[g(R0, . . . , Rn)v(Sn+ Xn+1); Mn< 0, Xn+1< −Sn]

= Ex[g(R0, . . . , Rn)v(Sn); Mn< 0] .

This consistency property allows (under suitable regularity conditions on the underlying probability space) the construction of probability measures P−_x, x ≤ 0 fulfilling for each n

E−_x[g(R0, . . . , Rn)] =

1

v(x)Exg(R0, . . . , Rn)v(Sn); Mn< 0 .

The transformation above is known as the Doob transform from the theory of Markov chains. Under P−_x the process S0, S1, . . . becomes a Markov chain with state space R− and transition kernel

P−

(x, dy) := 1

v(x)P{x + X ∈ dy}v(y)1{y<0} , x ≤ 0 .

As P−_{(x, [0, ∞)) = 0, the Markov process described by the transition matrix above never enters [0, ∞)}

again, although it may start from the boundary x = 0. Essentially, the transformation above describes a random walk conditioned to stay negative.

Similarly u gives rise to probability measures P+

x, x ≥ 0, characterized by the equation

E+_x[g(R0, . . . , Rn)] =

1

u(x)Ex[g(R0, . . . , Rn)u(Sn); Ln≥ 0] , n ∈ N0 . Under P+ _S

0, S1, . . . is a Markov process with state space R+0 and transition probabilities

P+_{(x, dy) :=} 1

u(x)P{x + X ∈ dy}u(y)1{y≥0}, x ≥ 0 . Intuitively, it is the random walk conditioned to stay nonnegative.

Remark. There is a slight difference between P+ and P−: under P+x the process S may hit 0, however,

under P−x this possibility is excluded. For x < 0 a difference only occurs for those x, where v(x) 6=

v(x−), that is for at most countably many x. If one considers (as below) measures P−

ν having an initial

distribution ν without atoms, there is no difference at all.

3.2.2 A conditional limit law

Here, we prove a conditional limit law for oscillating random walks. It is a generalization of [BD94, Theorem 1]. In the context of BPREs the ideas of the proof of the limit theorem can be found in [Vat04, Lemma 7]. We use similar arguments here, but in a more general context. The limit theorem is valid under a more general condition than Assumption 3.2, often referred to as Spitzer’s condition (see Chapter 2).

Assumption 3.5. There exists a number 0 < ρ < 1 such that 1 n n X k=1 P(Sk > 0) → ρ as n → ∞ .

The summands may also be replaced by P(Sk ≥ 0), as for every nondegenerated random walk,

Pn

k=1P(Sk = 0) = o(n) (see [Fel87, chapter XII]). Note that Assumption 3.2 implies 3.5.

We will need the asymptotic of P(τn = n). The next result follows from applying [AGKV05a, Lemma

Branching processes in random environment